$n$th- roots of unit in a splitting field

Question

Let $n \geq 2$ and $L$ be the splitting field of $x^n -1$ over $\mathbb{Q}$. How do I show that $[L : \mathbb{Q}] = \phi(n)$, the Euler function of $n$. I tried considering that the set of roots of unit is a group over multiplication in $L$ and showing that it is cyclical. How do I relate this question to roots of unit in the complex numbers?